copyright © 2011 pearson education, inc. time series chapter 27

Copyright © 2011 Pearson Education, Inc.

Time Series

Chapter 27

27.1 Decomposing a Time Series

Based on monthly shipments of computers and electronics in the US from 1992 through 2007, what would you forecast for the future?

Use methods for modeling time series, including regression.

Remember that forecasts are always extrapolations in time.


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The analysis of a time series begins with a timeplot, such as that of monthly shipments of computers and electronics shown below.


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Forecast: a prediction of a future value of a time series that extrapolates historical patterns.

Components of a time series are:

Trend: smooth, slow meandering pattern. Seasonal: cyclical oscillations related to

seasons. Irregular: random variation.


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Smoothing

Smoothing: removing irregular and seasonal components of a time series to enhance the visibility of the trend.

Moving average: a weighted average of adjacent values of a time series; the more terms that are averaged, the smoother the estimate of the trend.


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Smoothing

Seasonally adjusted: removing the seasonal component of a time series.

Many government reported series are seasonally adjusted, for example, unemployment rates.


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Smoothing: Monthly Shipments Example

Red: 13 month moving average Green: seasonally adjusted.


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Smoothing: Monthly Shipments Example

Strong seasonal component (three-month cycle).


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Exponential Smoothing

Exponentially weighted moving average (EWMA): a weighted average of past observations with geometrically declining weights.

EWMA can be written as . Hence, the current smoothed value is the weighted average of the current observation and the prior smoothed value.


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1)1( ttt wsyws


Exponential Smoothing

The choice of w affects the level of smoothing. The larger w is, the smoother st becomes.

The larger w is, the more the smoothed values trail behind the observations.


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Exponential SmoothingMonthly Shipments Example (w = 0.5)


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Exponential SmoothingMonthly Shipments Example (w = 0.8)


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27.2 Regression Models

Leading indicator: an explanatory variable that anticipates coming changes in a time series.

Leading indicators are hard to find.

Predictor: an ad hoc explanatory variable in a regression model used to forecast a time series (e.g., time index, t)


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Polynomial Trends

Polynomial trend: a regression model for a time series that uses powers of t as explanatory variables.

Example: the third-degree or cubic polynomial.


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tt ttty 33

2210


Polynomial TrendsMonthly shipments: Six-degree polynomial

The high R2 indicates a great fit to historical data.


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Polynomial TrendsMonthly shipments: Six-degree polynomial

The model has serious problems forecasting.


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Polynomial Trends

Avoid forecasting with polynomials that have high powers of the time index.


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4M Example 27.1: PREDICTING SALES OF NEW CARS

Motivation

The U.S. auto industry neared collapse in 2008-2009. Could it have been anticipated from historical trends?


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Motivation – Timeplot of quarterly sales (in thousands)

Cars in blue; light trucks in red.


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Method

Use regression to model the trend and seasonal components apparent in the timeplot. Use a polynomial for trend and three dummy variables for the four quarters.

Let Q1 = 1 if quarter 1, 0 otherwise; Q2 = 1 if quarter 2, 0 otherwise; Q3 = 1 if quarter 3, 0 otherwise.

The fourth quarter is the baseline category. Consider the possibility of lurking variables (e.g., gasoline prices).


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MechanicsLinear and quadratic trend fit to the data.

Linear appears more appropriate.


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MechanicsEstimate the model.

Check conditions before proceeding with inference.


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MechanicsExamine residual plot.

This plot, along with the Durbin-Watson statistic D = 0.84, indicates dependence in the residuals.

Cannot form confidence or prediction intervals.


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Message

A regression model with linear time trend and seasonal factors closely predicts sales of new cars in the first two quarters of 2008, but substantially overpredicts sales in the last two quarters.


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Autoregression

Autoregression: a regression that uses prior values of the response as predictors.

Lagged variable: a prior value of the response in a time series.


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Autoregression

Simplest is a simple regression that has one lag:

This model is called a first-order autoregression, denoted as AR(1).


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ttt yy 110


Autoregression Example: AR(1) for Monthly Shipments


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AutoregressionScatterplot of Shipments on the Lag

Indicates a strong positive linear association.


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Forecasting an Autoregression

Example: Use AR(1) to forecast shipments.

For Jan. 2008, use observed shipment for Dec. 2007:


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19706.09000.0ˆ tt yy

billionyJan 7385.31$ˆ 2008.


Forecasting an Autoregression

For Feb. 2008, there is no observed shipment for Jan. 2008. Use forecast for Jan. 2008:

Once forecasts are used in place of observations, the uncertainty compounds and is hard to quantify.


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billionyFeb 751.31$)785.31(9706.09000.0ˆ 2008.

27.3 Checking the Model

Autoregression and the Durbin-Watson Statistic

Example: Residuals from sixth-degree polynomial trend fit to monthly shipments plotted over time.


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Example: Residuals from sixth-degree polynomial trend fit to monthly shipments plotted over their lag.


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Residual plots show that the sixth-degree polynomial leaves substantial dependence in the residuals.

This dependence or correlation between adjacent residuals is known as autocorrelation (this first order autocorrelation is denoted as r1).


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The Durbin-Watson statistic is related to the autocorrelation of the residuals in a regression:


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)1(2)(

1

1

2

2

21

re

eeD n

ii

n

itt


Timeplot of Residuals

Useful for identifying outliers (e.g., April 2001).


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Summary

Examine these plots of residuals when fitting a time series regression:

Timeplot of residuals; Scatterplot of residuals versus fitted values; and Scatterplot of residuals versus lags of the

residuals.


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4M Example 27.2: FORECASTING UNEMPLOYMENT

Motivation

Using seasonally adjusted unemployment data from 1980 through 2008, can a time series regression predict the rapid increase in unemployment that came with the recession of 2009?


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Motivation


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Method

Use a multiple regression of the percentage unemployed on lags of unemployment and a time trend. In other words, use a combination of an autoregression with a polynomial trend.

The scatterplot matrix shows linear association and possible collinearity; hopefully the lags will capture the effects of important omitted variables.


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Mechanics

All conditions for the model are satisfied; proceed with inference.

Based on the F-statistic, reject H0. The model explains statistically significant variation. The fitted equation is


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)(164.0192.0794.0086.0ˆ 6121 ttttt yyyyy


Message

A multiple regression fit to monthly unemployment data from 1980 through 2008 predicts that unemployment in January 2009 will be between 7.02% and 7.66%, with 95% probability. Forecasts for February and March call for unemployment to rise further to 7.48% and 7.64%, respectively.


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4M Example 27.3: FORECASTING PROFITS

Motivation

Forecast Best Buy’s gross profits for 2008. Use their quarterly gross profits from 1995 to 2007.


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Method

Best Buy’s profits have not only grown nonlinearly (faster and faster), but the growth is seasonal. In addition, the variation in profits appears to be increasing with level. Consequently, transform the data by calculating the percentage change from year to year. Let yi denote these year-over-year percentage changes.


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MethodTimeplot of year-over-year percentage change.


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Mechanics

All conditions for the model are satisfied; proceed with inference.

The fitted equation has R2 = 71.0% with se = 7.37.

The F-statistic shows that the model is statistically significant. Individual t-statistics show that each slope is statistically significant.


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Mechanics

Forecast for the first quarter of 2008:

However, with se = 7.4, the range of the 95% prediction interval includes zero. It is [-6.5% to 25%].


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%345.9)282.11(383.0)318.0(443.0)285.2(911.0971.2ˆ y


Message

The time series regression that describes year-over-year percentage changes in gross profits at Best Buy is significant and explains 70% of the historical variation. It predicts profits in the first quarter of 2008 to grow about 9.3% over the previous year; however, the model can’t rule out a much larger increase (25%) or a drop (about 6.5%).


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Best Practices

Provide a prediction interval for your forecast.

Find a leading indicator.

Use lags in plots so that you can see the autocorrelation.


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Best Practices (Continued)

Provide a reasonable planning horizon.

Enjoy finding dependence in the residuals of a model.

Check plots of residuals.


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Pitfalls

Don’t summarize a time series with a histogram unless you’re confident that the data don’t have a pattern.

Avoid polynomials with high powers.

Do not let the high R2 of a time series regression convince you that predictions from the regression will be accurate.


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Pitfalls (Continued)

Do not include explanatory variables that also have to be forecast.

Don’t assume that more data is better.


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copyright © 2011 pearson education, inc. time series chapter 27

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